# Questions tagged [complex-manifolds]

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### Every closed orientable surface is Riemann surface

I want to prove that every closed orientable surface is a Riemann surface i.e. every closed orientable surface admits a complex structure. Several proofs are available which make use of classification ...
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### lower bound the distance between two varieties

$\DeclareMathOperator{\complex}{\mathbb{C}}$ Let $X,Y \subseteq \complex^n$ be homogeneous, smooth, irreducible, closed algebraic sets with $X \cap Y=\{0\}$. I would like to numerically lower bound ...
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### Holomorphic functions on Riemann surfaces with boundary

Suppose that $\Sigma$ is a compact Riemann surface with boundary and that $f: \Sigma \rightarrow \mathbb{C}$ is holomorphic*. If $f$ is real-valued along $\partial \Sigma$, is it necessarily true that ...
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### Why restrict to complex Lie algebras?

I am taking a class about Lie algebras, where we introduced in the beginning the notion of a Lie algebra, but over time we restricted ourselves only to complex Lie algebras. Can someone of you tell me ...
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### Showing that $\tau(t) = (t^2, t^3)$ is not a submanifold

Let $\tau : \mathbb{C} \to \mathbb{C}^2$ be the map $\tau(t) := (t^2, t^3)$. Show that $\tau$ defines an embedding map from $\mathbb{C}^*$ to $\mathbb{C}^2 \setminus{0}$. Is $\tau(\mathbb{C})$ a ...
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### Are the chart maps of a complex manifold necessarily biholomorphic?

I know that the transition maps of a complex manifold are biholomorphic, but are the chart maps themselves also biholomorphic? I know that it is the case for real smooth manifolds (here the chart maps ...
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### Dimension of the complex projective space $\mathbb{C}\mathbb{P}^n$ as almost complex manifold

I have already shown that the complex projective space $\mathbb{C}\mathbb{P}^n$ is a complex manifold by checking the required properties of the transition maps. Since every complex manifold is an ...
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### Holomorphic function spaces on Stein manifolds

This may be a long shot, but I'm interested in learning about holomorphic function spaces on Stein manifolds, however, I can't find much literature on the topic. I don't know if this is just too hard ...
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### Does uniformization theorem imply all 2d manifolds are confromally flat?

Here is a screenshot from nakahara. Now, to me it looks like that the uniformization theorem implies that all 2d manifolds are conformally flat, because the constant curvature metrics described in (14....
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### complex manifolds and geometry reference request

I am curious to learn about complex manifolds and complex geometry. I am familiar with the classical algebraic and analytic theory of Riemann surfaces, complex analysis in one variable (say, first ...
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### Manifold and hyperplane

Can someone explain me the relation between manifold and hyperplane. I saw a definition but I am not able to connect the idea. The definition is A set Γ ⊂ $R^n$ is called a k–dimensional $C_m$...
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### Singularities of moduli spaces $M_g$

This survey paper from Lizhen Ji says: Teichmüller was aware that nontrivial automorphisms of Riemann surfaces caused difficulty in constructing $M_g$ and singularities of $M_g$, and he ...
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### Definition of bubbles and Removal of singularities

In my lecture, I have the following theorem: Suppose $u:(B^2 \setminus \{0\}, i) \rightarrow (M,J)$ is $J$-holomorphic with $E(u)< \infty$ (energy) and such that the image of $u$ is contained in ...
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### Kummer suface ; the resolution has a holomorhic (2,0)-form .

At first, I am describing a resolution of Kummer surface: Get a lattice of rank 4 ; $\Gamma$ on $\mathbb{C}^2$. The quotient $\mathbb{T}^4:\mathbb{C}^2 /\Gamma$ would be a complex tori . Now ...
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### Line bundles on projective space and disk

I'm having a difficult time solving some exercice. I should prove the following : Show that any holomorphic line bundle on a disc $\Delta\subset\mathbb{C}$ is trivial. Deduce that any holomorphic ...
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### Show that $\log|f|$ is a plurisubharmonic function

$\Omega \subset C^n$. $f \in O(\Omega)$. Show that $\log|f|$ is a plurisubharmonic function. I have tried two methods. The first one is calculating Hessain matrix of $log|f|$, but it is too hard. ...
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### Show that the closure of $\cup_{k\geq1}F_k(D)$ is compact in $\Omega$

$\Omega$ is a domain of holomorphy, $D=\{z \in C : |z|\leq1\}$.For an arbitrary series of holomorphic functions $F_k:D\to \Omega$,The closure of $\cup_{k\geq1}F_k(\partial D)$ is compact in $\Omega$. ...
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### Fibre of smooth holomorphic map is manifold (ComGeo by Huybrechts)

I have a question on a remark from Daniel Hyubrechts' Complex Geometry Complex Geometry on page 107. Definition 2.6.13 A holomorphic map $f: X \to Y$ is smooth at a point $x \in X$ if the induced ...
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### How Quintic 3-fold is a Calabi–Yau manifold and has non vanishing Ricci scalar?

It’s well known that quintic 3-fold is a Calabi-Yau manifold in the complex projective space $\mathbb{CP}^{n+1}$ , see for instance: https://en.m.wikipedia.org/wiki/Quintic_threefold Now the main ...
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### Complex manifolds vs Riemann domains

In Hörmander's text an "Introduction to Several Complex Variables," he gives the following definition for a Riemann domain on page 139: A complex manifold $\Omega$ of dimension $n$ is called a ...
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### Sufficient Conditions for Deformations of a Complex Manifold to form an Almost-Quaternionic Structure

Let $(M, J)$ be a $2n$ (complex) dimensional manifold with complex structure $J$, and consider a differentiable family $\mathcal{F}=(\mathcal{M}, B, \phi)$ of complex structures with respect to $M$, ...
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### What does parametric disc mean in this context of triangulation?

Suppose I have two compact Riemann surfaces $\mathfrak{B}$ and $\mathfrak{B}'$ as well as a covering $f: \mathfrak{B}' \rightarrow \mathfrak{B}.$ My book says that I can triangulate the Riemann ...
Let $M$ be a complex manifold with a Kahler metric $g$, define the covariant derivative of a smooth complex valued $T^{(1,0)}M$ vector field $X = X^i\partial_i$ to be such that its i^{th} component is ...
I would expect that the dual of a vector bundle would be defined by the inverse conjugate transpose, as that would be the inverse of the adjoint. When $\alpha_{ij}:X\to Y$ is a transition matrix in $E$...